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Considering inviscid LNS equations:
\begin{eqnarray}
  \Gamma\frac{\partial \phi}{\partial t}
  +A\frac{\partial \phi}{\partial x}
  +B\frac{\partial \phi}{\partial y}
  +C\frac{\partial \phi}{\partial z}
  +D\phi=0
\end{eqnarray}
let $\phi=\tilde\phi(\xi, y, z)exp(\alpha x-\omega t), \xi=\epsilon x$, then
\begin{equation}
  -i\omega\Gamma\tilde\phi+i\alpha A\tilde\phi+A\frac{\partial\tilde\phi}{\partial x}
  +B\frac{\partial \tilde\phi}{\partial y}+C\frac{\partial\tilde\phi}{\partial z}+D\tilde\phi=0
\end{equation}
simplify it, it becomes:
\begin{equation}\label{eq3}
\hat D\tilde\phi+\hat A\frac{\partial\tilde\phi}{\partial x}
+\hat B\frac{\partial \tilde\phi}{\partial y}+\hat C\frac{\partial\tilde\phi}{\partial z}=0
\end{equation}
where,
\begin{eqnarray}
  &\hat D=-i\omega\Gamma+i\alpha A+D\\
  &\hat A=A\\
  &\hat B=B\\
  &\hat C=C
\end{eqnarray}
then, considering the wave is periodic in the spanwise direction, the shape function
can be written as
\begin{equation}\label{eq4}
  \tilde\phi=\sum_{m=-M}^{m=M}\hat\phi_m(\xi, y) exp(im\beta_0 z)exp(\gamma z)
\end{equation}
where, $\beta_0=2\pi/L_z, \gamma=\gamma_r+i\gamma_i, \gamma_r=0, \gamma_i=\epsilon \beta$.

we assume the basic flow operator
\begin{eqnarray}
  &\hat A=\sum_{m=-M}^{m=M}\hat A_m e^{im\beta_0 z}\\
  &\hat B=\sum_{m=-M}^{m=M}\hat B_m e^{im\beta_0 z}\\
  &\hat C=\sum_{m=-M}^{m=M}\hat C_m e^{im\beta_0 z}\\
  &\hat D=\sum_{m=-M}^{m=M}\hat D_m e^{im\beta_0 z}
\end{eqnarray}
and substitute (\ref{eq4}) into (\ref{eq3}), we can obtained:
\begin{equation}\label{eq6}
\begin{split}
&\sum_{m=-M}^{m=M}\hat D_m e^{im\beta_0 z}\sum_{m=-M}^{m=M}\hat\phi_m e^{im\beta_0 z}\\
&+\sum_{m=-M}^{m=M}\hat A_m e^{im\beta_0 z}\sum_{m=-M}^{m=M}\frac{\partial\hat\phi_m }{\partial x}e^{im\beta_0 z}\\
&+\sum_{m=-M}^{m=M}\hat B_m e^{im\beta_0 z}\sum_{m=-M}^{m=M}\frac{\partial \hat\phi}{\partial y}e^{im\beta_0 z}\\
&+\sum_{m=-M}^{m=M}\hat C_m e^{im\beta_0 z}\sum_{m=-M}^{m=M}{i(m+\epsilon)\beta_0\hat\phi}e^{im\beta_0 z}=0
\end{split}
\end{equation}
we find hat the nonlinear term: such as $\sum_{m=-M}^{m=M}\hat C_m e^{im\beta_0 z}\sum_{m=-M}^{m=M}{i(m+\epsilon)\beta_0\hat\phi}e^{im\beta_0 z}$ can be written as:
\begin{eqnarray}
&\sum_{m=-M}^{m=M}\hat C_m e^{im\beta_0 z}\sum_{m=-M}^{m=M}{i(m+\epsilon)\beta_0\hat\phi}e^{im\beta_0 z}=\sum_{m=-2M}^{m=2M} \bar C_me^{im\beta_0 z}\\
&\bar C_m=\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}i(l+\epsilon)\beta_0C_{m-l}\hat\phi_l
\end{eqnarray}
if we drop the truncation error in eq. (\ref{eq6}), then the equations can be written as:
\begin{eqnarray}
& \begin{split}
  &\sum_{m=-M}^{m=M}\bar D_m e^{im\beta_0 z}+\sum_{m=-M}^{m=M}\bar A_me^{im\beta_0 z}
  +\sum_{m=-M}^{m=M}\bar B_m e^{im\beta_0 z}+\sum_{m=-M}^{m=M}\bar C_m e^{im\beta_0 z}=0
  \end{split}\\
  &\bar D_m=\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\hat D_{m-l}\hat\phi_l\\
  &\bar A_m=\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\hat A_{m-l}\frac{\partial\hat\phi_l }{\partial x}\\
  &\bar B_m=\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\hat B_{m-l}\frac{\partial\hat\phi_l }{\partial y}\\
  &\bar C_m=\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}i(l+\epsilon)\beta_0C_{m-l}\hat\phi_l
\end{eqnarray}
Considering the orthogonality in the Fourier space, we can obtain that:

for $m=-M$ to $m=+M$,
\begin{equation}\label{eq21}
\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\hat D_{m-l}\hat\phi_l +\hat A_{m-l}\frac{\partial\hat\phi_l }{\partial x}+\hat B_{m-l}\frac{\partial\hat\phi_l }{\partial y} +i(l+\epsilon)\beta_0\hat C_{m-l}\hat\phi_l =0
\end{equation}
this equation describes the shapefunction $\hat \phi(xi, y, m)$.
we assume the shape function  $\phi$ is $\phi(i, j, m)$ with normalwise and streamwise discretization, then the equation can written as:
\begin{equation}
\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\hat A_{i, j, m-l}(c_2\phi_{i, j, l}+c_1\phi_{i-1, j, l})+(\hat D_{i, j, m-l}+i(l+\epsilon)\beta_0\hat C_{i, j, m-l})\hat\phi_{i, j, l}+\hat B_{i, j, m-l}\sum_{jj=j-2}^{j+2}d_{jj-j}\hat\phi_{i, jj, l} =0
\end{equation}
then, the equations is a linear system problem as below:
\begin{equation}
\begin{split}
&\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}(\hat A_{i, j, m-l}c_2+\hat D_{i, j, m-l}+i(l+\epsilon)\beta_0\hat C_{i, j, m-l})\hat\phi_{i, j, l}+\hat B_{i, j, m-l}\sum_{jj=j-2}^{j+2}d_{jj-j}\hat\phi_{i, jj, l} =\\
&-\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\hat A_{i, j, m-l}(c_1\phi_{i-1, j, l})
\end{split}
\end{equation}
then, we consider the viscous terms in the LNS equations. It means that the equation is :
\begin{eqnarray}
  \Gamma\frac{\partial \phi}{\partial t}
  +A\frac{\partial \phi}{\partial x}
  +B\frac{\partial \phi}{\partial y}
  +C\frac{\partial \phi}{\partial z}
  +D\phi=
  V_{xx}\frac{\partial^2 \phi}{\partial x^2}
  +V_{xy}\frac{\partial^2 \phi}{\partial xy}
  +V_{xz}\frac{\partial^2 \phi}{\partial xz}
  +V_{yy}\frac{\partial^2 \phi}{\partial y^2}
  +V_{yz}\frac{\partial^2 \phi}{\partial yz}
  +V_{zz}\frac{\partial^2 \phi}{\partial z^2}
\end{eqnarray}

then, with the viscous terms, the equations (\ref{eq21}) become that:
\begin{equation}
\begin{split}
&\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}(\hat D_{m-l}+\alpha^2\hat V_{xx,m-l})\hat\phi_l \\
&+(\hat A_{m-l}-i\alpha \hat V_{xy,m-l}-i\frac{\partial \alpha}{\partial x}\hat V_{xx,m-l})\frac{\partial\hat\phi_l }{\partial x}+(\hat B_{m-l}-i(l+\epsilon)\beta_0 \hat V_{yz, m-l})\frac{\partial\hat\phi_l }{\partial y}\\
&+(i(l+\epsilon)\beta_0\hat C_{m-l}+(l+\epsilon)^2\beta_0^2\hat V_{zz,l-m}+\alpha(l+\epsilon)\beta_0 \hat V_{xz,m-l})\hat\phi_l\\
&-\hat V_{yy,l-m}\frac{\partial^2\hat\phi_l}{\partial y^2} =0
\end{split}
\end{equation}
after the similification, the equation is:
\begin{eqnarray}
&\begin{split}
&\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\tilde A_{m-l}\frac{\partial\hat\phi_l }{\partial x}+\tilde B_{m-l}\frac{\partial\hat\phi_l }{\partial y}+\tilde D_{m-l}\hat\phi_l+\tilde H_{m-l}\frac{\partial^2\hat\phi_l}{\partial y^2} =0
\end{split}\\
&\tilde A_{m-l}=\hat A_{m-l}-i\alpha \hat V_{xy,m-l}-i\frac{\partial \alpha}{\partial x}\hat V_{xx,m-l}\\
&\tilde B_{m-l}=\hat B_{m-l}-i(l+\epsilon)\beta_0\hat V_{yz, m-l}\\
&\tilde D_{m-l}=\hat D_{m-l}+\alpha^2\hat V_{xx,m-l}+i(l+\epsilon)\beta_0\hat C_{m-l}+(l+\epsilon)^2\beta_0^2\hat V_{zz,l-m}+\alpha(l+\epsilon)\beta_0 \hat V_{xz,m-l}\\
&\tilde H_{m-l}=-\hat V_{yy,m-l}
\end{eqnarray}
with normalwise and streamwise discretization, the above equations become a linear system, i.e.:
\begin{eqnarray}
&\begin{split}
&\sum_{l=-min(M-m, M)}^{l=min(M+m, M)} (\tilde A_{i, j, m-l}c_2+\tilde D_{i,j, m-l})\hat\phi_{i,j,l}+\sum_{jj=j-2}^{j+2}(d_{jj-j}\tilde B_{i,j, m-l}+e_{jj-j}\tilde H_{i,j, m-l})\hat\phi_{i,jj,l} =\\
&\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\tilde A_{i, j, m-l}(c_1\hat\phi_{i-1, j, l})
\end{split}\\
&\tilde A_{m-l}=\hat A_{m-l}-i\alpha \hat V_{xy,m-l}-i\frac{\partial \alpha}{\partial x}\hat V_{xx,m-l}\\
&\tilde B_{m-l}=\hat B_{m-l}-i(l+\epsilon)\beta_0 \hat V_{yz, m-l}\\
&\tilde D_{m-l}=\hat D_{m-l}+\alpha^2\hat V_{xx,m-l}+i(l+\epsilon)\beta_0\hat C_{m-l}+\alpha(l+\epsilon)\beta_0 \hat V_{xz,m-l}+(l+\epsilon)^2\beta_0^2\hat V_{zz,l-m}\\
&\tilde H_{m-l}=-\hat V_{yy,m-l}
\end{eqnarray}
For a 2D eigenvalue problem(2D-EVP), the disturbance here $\phi=\tilde\phi(y, z)exp(\alpha x-\omega t)$. It means that the shape function $\tilde\phi$ has no relationship with the streamwise coordination. All the terms contains the streamwsie variation can be ignored. Then, the equation is an 2D EVP. It is:
\begin{eqnarray}
&\begin{split}
&\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}\tilde D_{m-l}\hat\phi_{i,j,l}+\sum_{jj=j-2}^{j+2}(d_{jj-j}\tilde B_{i,j, m-l}+e_{jj-j}\tilde H_{i,j, m-l})\hat\phi_{i,jj,l} =\sum_{l=-min(M-m, M)}^{l=min(M+m, M)}i\omega\hat\Gamma_{i,j,m-l}
\end{split}\\
&\tilde B_{m-l}=\hat B_{m-l}-i(l+\epsilon)\beta_0 \hat V_{yz, m-l}\\
&\tilde D_{m-l}=\hat D_{m-l}+i\alpha\hat A_{m-l}+\alpha^2\hat V_{xx,m-l}+i(l+\epsilon)\beta_0\hat C_{m-l}+\alpha(l+\epsilon)\beta_0 \hat V_{xz,m-l}+(l+\epsilon)^2\beta_0^2\hat V_{zz,m-l}\\
&\tilde H_{m-l}=-\hat V_{yy,m-l}
\end{eqnarray}
\end{document}
